A simple proof of Barr's theorem for infinitary geometric logic (joint work with Roy Dickhoff)

Abstract:

Geometric logic has gained considerable interest in recent years: contributions and applications areas include structural proof theory, category theory, constructive mathematics, modal and non-classical logics, automated deduction, and even Kantian philosophy. Geometric logic studies the so-called geometric or coherent theories, i.e. first-order theories axiomatized by geometric or coherent implications. Several mathematical theories--such as the theory of torsion abelian groups, of Archimedean ordered fields, and of connected graphs--are axiomatized by infinitary geometric implications. Also notions such as the transitive closure of accessibility relations, essential in the semantical analysis of aggregation of epistemic attitudes, are expressed in infinitary geometric logic.

Proof analysis is extended to all such theories by augmenting an infinitary classical sequent calculus with a rule scheme for infinitary geometric implications. The calculus is designed in such a way as to have all the rules invertible and all the structural rules (weakening, contraction, and cut) admissible. An intuitionistic infinitary multisuccedent sequent calculus is also introduced and it is shown to enjoy the same structural properties. Finally, it is shown that by bringing the classical and intuitionistic calculi close together, the infinitary Barr theorem becomes an immediate result.

Concepts, Proto-concepts, and Artificial Intelligence(joint work with Salvatore Gaglio)

Abstract:

In "Wittgenstein, Turing, and Neural Networks" (WTNN) we argued, among other things, that one of the corner stones of analytical philosophy, the so-called 'priority thesis' cannot be right. Let us mention, in passing, that, according to the priority thesis, the study of language is prior, in the order of explanation, to the study of thought.This paper is a follow-up to WTNN focussing on aspects of conceptual activity that have to do with the ability to make distinctions. In particular, after having, preliminarily, drawn a demarcation line between concepts and proto-concepts, we proceed by discussing some stimulus-response cognitive systems (Convolutional Neural Networks) that can be trained to make distinctions typical of proto-concepts in the absence of high-level cognitive functions such as consciousness, understanding, representation, and intentionality.

We believe that such cognitive systems (CNNs) might cast light on a region of conceptual activity which, far from being the ultimate product of a `linguistic mind,' is, rather, inscribed in the nuts and bolts of these systems' biology/electronics.

Trivalent Semantics for Indicative Conditionals (joint work with Paul Égré and Jan Sprenger)

Abstract:

The indicative conditional "if A, then C" has the form of a declarative sentence; yet it is hard to define satisfactory truth conditions for it within bivalent (classical or modal) logic. In this paper, we explore trivalent truth conditions as an alternative: indicative conditionals are understood as conditional assertions that take the value of its consequent whenever its antecedent is true, and the value indeterminate whenever its antecedent is false. This semantics has already been proposed by Bruno de Finetti in the 1930s, but the resulting logics have barely been studied. First, we investigate the benefits of de Finetti's approach, analyze different truth tables for the indicative conditional, and combine them with suitable notions of validity. We argue that the most promising trivalent logic for the indicative conditional is based on the truth table proposed by Cooper (1968) and Cantwell (2008), paired with a tolerant-to-tolerant (or TT-)notion of validity. Second, we focus on the proof theory of the different logics (offering sound and complete tableaux and sequent calculi) and on their algebraization.

The following presentation is meant to provide a general overview of the philosophical debate about the principle of least action (PLA). Firstly, i will introduce the Lagrangian formalism and its relation to PLA. Secondly, i will analyse some of the most popular philosophical interpretations of such principle (i.e. dispositions, teleology and modality). Ultimately, I present PLA as emergent from the Feynman path integrals where the latter are to be taken as an holistic ensemble of possible paths.

Many philosophers are familiar with the doctrine of the "open future" - the doctrine, roughly speaking, that claims about undetermined aspects of the future currently fail to be true. For instance: it is not now true that it will rain tomorrow, but it is also not now true that it will /not / rain tomorrow. The future is, in this sense, "open". The open future doctrine has always provoke dcontroversy on both metaphysical and logical grounds; in particular, must the open futurist deny the classical principle of bivalence? In this talk, I aim to make progress on these difficult questions by investigating what we might say about a different sort of openness, a sort of openness that almost no one accepts - the openness of the past. I defend a picture of the open past - and the open future - that preserves classical logic.

The theory of repeated games asserts that when past conduct is unobservable, patient individuals can cooperate if defections impose large losses on cooperators, and if everyone sanctions a deviation by defecting forever (Kandori, 1992). Here we show that this extreme “grim” punishment is not necessary and, in fact, may be counterproductive if individuals are sufficiently patient. We prove that a class of moderate punishments exists, which has the advantage of supporting full cooperationwithout having to arbitrarily restrict off-equilibrium payoffs. Our theory provides a rationale for the empirical observation that grim punishment is uncommon in laboratory studies of cooperation.

Abstract: Essentially all popular contemporary accounts of the semantics of scientific theories bases meaning on something like truth conditions in something like a Tarskian sense: I know the meaning of a scientific theory when I know how to interpret its terms using designation relations such that its propositions come out true. What was wanted was an account of "meaning" analytically connected to truth, completely divorced from human concerns. But this is self-defeating, for such a conception completely divorces semantics from the fundamental sources of scientific knowledge ---experimental knowledge--- which in the end must ground the empirical content and significance of our theoretical representations. An adequate account of semantics, I argue, will return to the Carnapian conception that semantics must ground analysis ofepistemology, and be grounded in turn by our grasp of it. Thus, an account of semantics must make explicit links to scientific knowledge in all its human forms: as achieved state, both in theory and in experiment; as mediator of evidentiary relations and provider of epistemic warrant; and as ground for the successful continuation of thescientific enterprise, the extension and deepening of the first kind by application of the second.

Workshop On Predication

venerdì 23 febbraio 2018, ore 9:45

9:45 Welcome

10:00 Øystein Linnebo (University of Oslo)Ontological categories and the problem of expressibility (joint work with Bob Hale)

Abstract:Frege famously held that the ontological categories correspond to the logico-syntactic types. Something is an object just in case it can be referred to by a singular term, and likewise for all the other categories. This view faces an expressibility problem. In order to express the view, we need to generalize across categories; but by the view itself, any one variable can only range over a single category. We provide a sharp formulation of the problem, show that there is no easy way out, and then explore some of the hard ways.

Abstract: The relations between Frege’s notions of concept and of object are analyzed from a logical point of view. Such relations turn out to go beyond that of falling under, so the standard theory must be widened. The result is a theory of a general kind of relations that we call copular relations. Arguments are given to show that this notion of copularity is logically founded and not merely conventional.

13:00 Lunch

14:00 Fraser Macbride (University of Manchester)Relations: predicates expressing them and names denoting them

Abstract:I argue that predicates in general and many place predicates in particular are impurely referring expressions, i.e. do not only refer to relations but perform a further co-ordinating function in virtue of which a sentence is more than a list. Conceiving of predicates as impurely referring expressions not only provides a solution to Frege's Paradox of the Concept Horse but also allows us to address the Puzzle about Relation Names advanced by van Inwagen. Because it enables us to solve these puzzles, this gives us reason to favour my view that predicates are impurely referring expressions.